Map Making for Social Scientists

American Sociological AssociationAnaheim, 20 August 2001

Waldo ToblerProfessor Emeritus of Geography

University of California at Santa Barbarahttp://www.geog.ucsb.edu

Some hot topics in contemporary cartography

Animation of geographical objects

Three dimensional visualization

Map making on the internet

Map generalization

I will emphasize three other subjects

Map projections

Dealing with aggregate dataSpatial filteringEstimating densitiesConverting to other units

Depicting movement

First, very quickly, map projections

The mapping processCommon Surfaces Used in Cartography

The surface of the earth is two-dimensional

Sphere or Ellipsoid?

The departure of the earth from a sphere is approximately one part in three hundred

This is 3/10ths of one percent

This can be used as a rule of thumb:Is your work accurate to better than one percent?

Sphere or Map?

This is equivalent to asking whether you want to work in latitude and longitude or in plane coordinates

Programs exist, for example, to convert from street address to lat/lon. There are also programs to convert from lat/lon to X, Y, and visa versa

Many kinds of analysis are very simple on a sphereThis includes such things as distance, direction, or area

computationA plane is a sufficiently good approximation to a sphere

for a small areaWhat is small?

You can glue a postage stamp, without wrinkling it, on a 20 cm globe

Many analytical problems can be solved directly in geographic coordinates

This is often easy when the earth is considered spherical

It is more difficult to work in ellipsoidal coordinates

Some people like to work in plane, Euclidean, coordinates. Then a map projection is neededOf course the projection must be suited to the

problem, and there are many choices

Plane Coordinate Systems Are Based on Map Projections

The two most important ones are

The Universal Transverse Mercator systemThe State Plane Coordinate system

The equations for both are complicated and based on an ellipsoid

Virtually all countries of the world have similar systems

All map projections result in distorted maps!

Since the time of Ptolemy the objective has been to obtain maps with as little distortion as possible

Most geographic information systems and government mapping agencies take this point of view

But then Mercator changed this by introducing the ideaof a systematic distortion to assist in the solution of a problem

Mercator’s famous anamorphose helps solve a navigation problemHis idea caught on

Anamorphic projections are used to solve problems and are not primarily for display

One way to use map projections

It is useful to think of a map projection like you are used to thinking of graph paper

Semi logarithmic, logarithmic, probability plots, and so on, are employed to bring out different aspects of

data being analyzed

Map projections may be used in the same way

This is not a common use in geographic information systems

Hägerstrand’s Logarithmic Map

A map projection to solve a special problem

The next illustration shows the U.S. population assembled into one degree quadrilaterals

We would like to partition the U.S. into regions containing the same number of people

There follows a map projection (anamorphose) that may be useful for this problem

US Population By One Degree Quadrilaterals

Now Use the

Transform-Solve-Invert Paradigm

Transform the graticule, and map, to obtain areas of equal population

Then position a hexagonal tesselation on the map

Then take the inverse transformation

W. Tobler, 1973, “A continuous transformation useful for districting”, Annals, N.Y Academy of Sciences, 219:215-220

Next topicOften we deal with data given by areal units

Such as census tracts, counties, states, or other administrative units

It is convenient to think of the data as being binned into these spatial units in a manner similar to the

making of histogramsThe difference is that the bins are of irregular sizes,

shapes, and orientation on the surface of the earth

A choropleth (area filling) mapwith shading proportional to density

The same data shown as a bivariate geographic histogramwith bin heights proportional to density

I will consider three problems relating to such binnings

1. The filtering of data in the irregular spatial unitsincluding map generalization

2. Converting to continuous densities

3. Converting between areal units

Spatial filtering typically uses nearby, local, observations

Processing using neighbors is common in image processing. The value in a cell is converted to a weighted average of the

values in neighboring cells.Depending on the weights one obtains either smoothing

(a.k.a. blurring) or sharpening.Local geographic measures are similar in that they compute a

value at each location that depends on nearby values. There are many examples.

Modifying the center cell in the case of pixelsNeighborhood operators are used frequently in image processing

Neighborhood Operators Can Also Be Used With ReselsFirst and Second Order Neighbors of Kansas

Choropleth map of university attendanceAdrian Herzog, Zürich

University attendance, adjustedAdrian Herzog, Zürich

Unemployment, June 2001, by countyUSA Today, 20 August 2001, page 4B

US unemployment map, two detail viewsBrown: < 3.3%, Tan: 3.3-4.4%, Green: 4.5-6.2%, Red: >6.3%

Now a word about resolution

Average resolution can be calculated as

(area of domain / number of observations)1/2

In three dimensions use the cube root

In effect this measures the average distance influence of each observation

Unequal resolution in different parts of a map has an effect similar to unequal

magnification in a microscope

Average resolution ~55 km. Patterns >110 km detectableIn these resels the resolution varies across the US. Patterns within cities cannot be seen

Social data are often made available in a hierarchy of administrative units

Moving up through the hierarchy changes the resolution and this acts as a low pass spatial filter

The result is a less detailed - more blurred - map

Consequently I recommend using the finest data available

For example

The Dutch administrative hierarchy

Swiss migration at reduced resolutionTo emphasize the filtering effect of resolution

Another type of map generalization

Courtesy of Dr Guido Dorigo, University of Zurich

14.7 km resolution (184 Districts) 39.2 km resolution (26 Cantons)

Three levels of administrative units and three levels of migration resolution all at once.

Communities Districts Cantons

An across boundary problemCourtesy of Dr. Claude Grasland, Paris

In order to “uniformize” the resolution the bins in France are aggregated up the political

hierarchy

They then more nearly match the resolution of the Belgium information.

Had this not been done the resulting density for France would appear to have much more variability than that of

Belgium.

But this variability would be an artifact of the difference in resolution.

Population along the French-Belgium borderCourtesy of Dr. Claude Grasland, Paris

Conversion From Areal Units to Densities

A Gaussian kernel function

The data values are assigned to the centroid of the administrative units and then summed using weights taken from a sliding kernel function.

Exponential kernels are also used.

How this worksPosition the chosen kernel on the map

Search for all centroids within the kernelPick a weight from the kernel depending on the

distance of the centroid from the map locationMultiply the value at a centroid by the kernel weight

Sum all of the weighted values within the kerneland assign this value to the location of center of the

kernelMove to the next location and repeat

After all locations have been evaluated you are done and can contour the results

Density based on a Gaussian kernel with a 5 km span

Using a Gaussian kernel with a 10 km span

Three references for further reading on density estimation techniques

D. Scott, 1992, Multivariate Density Estimation, J. Wiley, New York.

B. Silverman, 1984, Density Estimation for Statistics and Data Analysis, Chapman & Hall, New York.

R. Tapia & I. Thompson, 1978, Non-Parametric Probability Density Estimation, Baltimore, Johns Hopkins U. Press.

Kernels can also be applied to dot maps

Each dot is assigned a value of one unit(dots with numerical values can also be used)

The distance of each dot from the center of the kernel is calculated

Then the dot values are modified by the kernel weightThe weighted values within the kernel are summed

and assigned to the location of the kernel centerThe map is complete when the sum has been calculated for all

locationsThus the dot distribution has been converted to a density map

A uniform kernel is often usedbut is not recommended because of its effects

This kernel inverts some peaks and valleys. See: J.Holloway, “Smoothing & Filtering of Time Series & Space

Fields”, Advances in Geophysics, 3 (1958): 351-389

There is also a method that avoids the use of kernel functions

It is sometimes referred to as areal interpolation.From this point of view it is incorrect, in my opinion,

to assign areal observations to points (centroids).One criterion to be satisfied is that the resultant

maintain the data values within each unit.The method is known as pycnophylactic reallocation.

Pycnophylactic Reallocation(Mass Preserving)

Allows the production of density or contour maps to be made from areal data.

It is reallocation - and somewhat of a disaggregation operator. My assertion is that it may actually

improve the data.It is also important for the conversion of data from one

set of statistical units to another, as from census tracts to school districts.

1st example

Population density in Kansasby county

Courtesy of T. Slocum

A piecewise continuous surface

Population density in Kansasafter mass preserving reallocation

Each County Still Contains the Same Number of People

A smooth continuous surface, with population pycnophylactically redistributed

Another example

Migration from Illinois shown first as a piecewise continuous bivariate geographical histogram, based on state outlines, with volumes according to Illinois outmigration

Recall that most migrants in Illinois relocate within the state

The same data is then shown as pycnophylactically interpolated

The smoothed surface can be partitioned to yield estimated migration by arbitrary regions - the

Great Lakes basin for example

Bivariate histogram of Illinois outmigration by state

Illinois outmigration pycnophylactically smoothed

Another example

This time using population data by Federal Planning Regions for Germany.

First the data are represented in a perspective view of a bivariate geographical histogram.

This is followed by a similar view of the continuous population density distribution.

Wolf-Dieter Rase, 2001, “Volume-preserving interpolation of a smooth surface from polygon-related data”, J. Geograph. Syst, 3:199-213.

How pycnophylactic reallocation works

Philosophically it is based on the notion that people are gregarious, influence each other, are

mobile, and tend to congregate.This leads to neighboring and adjacent places

being similar.Mathematically this translates into a smoothness

criterion (with small partial derivatives).

It applies to any data exhibiting spatial autocorrelation.

Left to Right

1. Data polygons 2. Rasterized 3. Smoothed

How the smoothing is done.

Imagine that each unit is built up of colored clay, with a different color for each unit.

The volume of clay represents the number of people, say, and the height represents the density.

In order to obtain smooth densities a spatula is used to smooth the surface, but no clay is allowed to

move from one unit into another. Color mixing is not allowed.

This, converted to mathematics, is what the computer program does.

Density from dot maps without using kernels

The pycnophylactic method can also be used to prepare smooth density maps from data given at

spot locations.Step 1. Use the inverse area of Dirichlet (a.k.a. Thiessen) regions as the density for each location.

If weights are attached to the locations divide these by the region area.

Step 2. Smooth the resulting densities by the pycnophylactic reallocation method.

Another important advantage of mass preserving reallocation

A frequent problem is the reassignment of observations from one set of collection units to a different set, when the

two sets are not nested nor compatible. For example converting the number of children observed by census tract to a count by school district. Area boundaries also usually change over time, requiring reallocation for compatibility.

The density values obtained using the smooth pycnophylactic method allow an estimate to be made rather

simply. A “cookie cutter” can cut the continuous (clay) surface into the new zones with subsequent addition

(summation) to get the count.

The last topic is the depiction of geographical movement

A great deal of change in the world is due to geographical movement

Movement of information, of people, of money, or of material

Animation is well suited to depicting this dynamic cartography

Tables are an important way of recording data on geographic movement

Especially when the rows and columns refer to known geographic locations

The tables are then “square”, having the same number of rows as columns

The entries in the tables record the amount of movement during some period of time

Such tables can be decomposed into two parts, a symmetric part and a skew symmetric part

For the statisticians in the audience the total variance can also be partitioned into these two parts

From B to A is not the same as A to B(Gary Larson)

An example

In the United States the currency indicates where it was issued

For bills this is the Federal Reserve District.Coins contain a mint abbreviation.

You can check your wallet to estimate your interaction with the rest of the country.

Dollar Bill(Federal Reserve Note)

The 12 Federal Reserve Districts(Alaska and Hawaii Omitted)

Movement of One Dollar NotesBetween Federal Reserve Districts, in hundreds, Feb. 1976

To: B NY P Cl R A Ch SL M K D SF

From: BostonNew York

PhiladelphiaClevelandRichmond

AtlantaChicago

St. LouisMinneapolisKansas city

DallasSan Francisco

The table of dollar bill movements

was obtained from MacDonalds outlets throughout the United States.

Source: S. Pignatello, 1977, Mathematical Modeling for Management of the Quality of Circulating Currency, Federal

Reserve Bank, Philadelphia

From the table we can compute a movement map.

Dollar Bill Movement in the U.S.

The map is computed using a continuous version of the gravity model

The result is a system of partial differential equations solved by a finite difference iteration to obtain the

potential field.

This can be contoured and its gradient computed and drawn on a map.

W. Tobler, 1981,"A Model of Geographic Movement", Geogr. Analysis, 13 (1): 1-20G. Dorigo, & Tobler, W., 1983, “Push Pull Migration Laws”, Annals, AAG, 73(1):1-17.

First the Federal Reserve Districts are “rasterized”

There will be one finite difference equation for each node on this raster(2088 simultaneous equations)

Solving the equations yields the potentialshown here by contours

The raster is indicated by the tick marks. The arrows are the gradients to the potentials. The streakline map is obtained by connecting the gradient vectors.

The same technique can be applied to other types of movement

For example the migratory movement of people.

Nine Region Migration TableUS Census 1965-1970

(Note asymmetry. There are places of depletion and accumulation.)

This is an example of a census migration table. There are also (50 by 50) state tables and county by county tables.

There is a great deal of spatial coherence in the migration pattern

Choropleth maps do not show this clearly.

In the U.S. case the state boundaries hide the effect. Therefore a clearer picture emerges if they are omitted.

There is also temporal coherence.

W. Tobler, 1995, “Migration: Ravenstein, Thornthwaite, and Beyond”, Urban Geography,16(4):327-343.

Gaining and Losing StatesBased on the marginals of a 48 by 48 migration table

State centroids used with symbol magnitude proportional to the amount of change

The conventional net movement mapBased on movement between state centroids

(Computer sketch. Optimum deletion: values below mean ignored)

This information can be converted to a potential field and its gradient

For this a model is required.

The model is, in essence, a continuous version of the familiar gravity model.

The gradients can also be connected to give a streakline map.

The next maps are based on the same observations as the previous map.

The pressure to move in the USA continuous spatial gravity model from a movement table

Recall that several million people migrate during the 5 year census period

The next map shows an ensemble average,not the path of any individual.

But observe, not unrealistically, that the people to the East of Detroit tend to go to the Southeast, and

Minnesotans to the Northwest, and the remainder to the Southwest.

Migration potentials and streaklinesThe streaklines are drawn by connecting the gradients to the potentials

By the insertion of arbitrary areal boundaries, and by measuring the amount of flux across these boundaries, one can obtain information not contained in the original data,

i.e., make a prediction.

It’s like using a cookie cutter pressed into the continuous flow model to look at an arbitrary piece and computing

the flow across its borders.

The next map is an example, using state boundaries.The US Census Bureau does not provide this information.

The model is used to make the prediction

Major Flux Across State BoundariesPredicted from the model and table marginals

If we used the 3,141 counties of the United States the migration table could contain

9,862,740 numbers

This is not a lot for a computer, but for humans?

We need models and visualization techniques!

Cartography provides excellent visualization and always requires a model.

To concludeI have emphasized three topics

Map projections

Dealing with aggregate dataSpatial filteringEstimating densitiesConverting to other units

Maps of movement

Thank you for your attention

For more detailed information go tohttp://www.geog.ucsb.edu/~tobler/presentations

andhttp:/www.geog.ucsb.edu/~tobler/publications/reprinted articles

Waldo Tobler